Rosario Dawson is not like other Hollywood actors. Consider this: she's 32, and in her 20s decided she'd had enough of being judged on her looks, so took to wearing enormous sweatshirts to auditions.

"I'd perform my ass off, and the casting directors would be like, 'You are perfect for this role, but can you wear something a little less shapeless?'" Her manager would bargain with her. She could wear a roll-neck jumper, he said – but could it at least be a fitted one? "I'm like, 'Ugh, fine', but these stupid conversations needed to be had, because unfortunately, don't believe what they tell you, there's very little imagination in Hollywood." She hoots with laughter.

It annoyed her when casting directors asked to see her in more revealing clothes, she says, because she was naked in the film Alexander, "so go to any crazy, sick website and you'll be able to look at it in slow motion if you like". Does that bother her? "No, not at all, my point being: then don't complain, 'We don't know what she really looks like.' Are you kidding?! Do your research. 'She looks a little fat right now'," she says, recalling a message that filtered down from some rotten, deluded film executive. "Really? They're called breasts … There was definitely a period for a couple of years where I rebelled against it. It probably cost me a lot of really big jobs, but I was just so angry."

I had been worried Dawson would be too tired to talk properly. Earlier in the day, she had called to put the interview back two hours, pleading jetlag, her voice full of mid-Atlantic grogginess. But she arrives at the Guardian on foot, poses quickly for a photo, sits down and she's away, words tumbling out.

She's been a women's activist for years, and I realise how steeped she is in feminist argument when she talks about how public-sector cuts are affecting women in the UK. (Dawson has a flat in London, but this still takes me aback.) She's active in all sorts of ways – she's a long-time volunteer with a girls' club where she grew up in Manhattan, and appears in the feminist documentary Miss Representation. Later this month she's performing in A Memory, a Monologue, a Rant and a Prayer, a benefit in London for the organisations V-Day and Women for Women International. The event is based on writings about violence against women, edited by playwright, activist and close friend, Eve Ensler.

The piece Dawson is performing is radical. Written by Periel Aschenbrand, In Memory of Imette starts with the narrator being terrified by the murder of a female student near her apartment. She arms herself with weapons including "a big-ass hunting knife," Aschenbrand writes, "with which, if need be, I could slice off someone's testicles". I ask how Dawson feels about performing the monologue, and she says she completely agrees with its central message, that men and women need to talk more about rape. "You know, don't just walk down the street and be like everything's peaches and roses. It's one in three women who are going to be raped, killed, beaten or abused in her lifetime, and that's just real. To not live with that as a reality is really dangerous for women, and it lets a lot of guys off the hook from really paying attention to what's happening to the women around them. Because it's not all the men who are doing it, but not every single guy that boasts in the locker-room about the hot sex he had last night, had it with someone who was conscious."

It's not the first time Dawson has addressed the subject of rape head-on. She produced the 2007 film Descent, written and directed by her friend Talia Lugacy, and starred as Maya, a student who is raped by a classmate. The character goes on to exact revenge, in one of the more extreme scenes in modern, mainstream film-making, but the story is also thoughtful. It shows the slow arc of Maya's brutalisation, and her feelings after striking back, too.

Being a producer on the film provided some useful distance, she says.

"Otherwise I could have disappeared into that character more, you know, and it would have taken me down. It was really depressing … But I thought it was important to show and really talk about revenge, and to put that question into people's minds. People have all these ideas about it, but what it would actually look like is not a triumph. It's actually really degrading and sad." After the film came out, Ensler invited her to sit on the board of V-Day, a movement to end violence against women. "I remember exactly where I was when she asked," she says, "and I was so excited."

Dawson's career has taken her through gritty dramas (He Got Game), broad teen comedies (Josie and the Pussycats), musicals (Rent), very broad adult comedies (Clerks II) and children's films (Zookeeper). It includes the comic book fantasy, Sin City – a project that reflects her lifelong love of comics. (In 2006, she co-created her own comic series, Occult Crimes Taskforce.)

She started out playing Ruby in Larry Clark's 1995 film Kids, aged 15. Written by Harmony Korine, Kids is a tough, troubling film, opening with a scene of child sex and moving through drugs, theft, extreme violence, racism, rape and brutal conversations about men having sex with disabled women. In its midst, Dawson seemed one of the few mild beacons of hope. Her character was tough, too, laughing and joking about the difference between sex, making love and fucking (she preferred the last), but there was something essentially redemptive about her.

Although she's very different to that character, she understood her circumstances, having grown up on New York's Lower East Side herself.

Her mother was 17 when Dawson was born, and only found out she was pregnant when she was picked for the 1980 Olympic volleyball team and had to take a test. (The US Olympics team boycotted that year for political reasons, so it didn't affect her participation.)

Dawson's biological father was not around, but when her mother was eight months' pregnant she started seeing a man she'd known for years, who went on to adopt her daughter. "I think about that now," says Dawson, "such a young man, marrying a woman with a baby who's not his – that just doesn't happen. He just loved my Mom, and he loved me, and I loved my Dad, you know?"

She's never met her biological father. "I tried looking him up online, and 70-something names showed up, some of them only with addresses, and I thought: I'm not going to do that … Maybe if I have a child, I'll want to know, just for medical history reasons." She was "violently afraid" of becoming a teenage mother herself, aware of how it had limited her mother's options, but the experience of being adopted has made her keen to follow that lead – ideally to adopt an older child, who's otherwise unlikely to find a home.

When she was growing up, Dawson's father worked in construction, and her mother did a variety of jobs – electrician, plumber, typist – but the family faced financial straits. They lived, initially, "in this slumlord apartment, with rats, tilted floors, a bath tub in the kitchen". There was a farmers' market nearby and her mother "used to get food out of the bins. It was fresh food, but technically speaking, she was bin-diving. We still ate and we were eating organic," she gives a wry smile. "But that's a pretty tough thing as a Mom to have to do."

They moved into a squat when she was six and her brother Clay was one. "A place with a huge, gaping hole in the ground and plastic for windows. I saw the stress on my parents. We were the only children in the building for years, because no one else was that crazy. But we had a wonderful childhood because of it. Everybody who moved in had different apartments, and it wasn't until the sewage lines and the electricity went in that everybody disappeared behind their doors. People really needed each other beforehand."

Her mother was always an activist; when Dawson was 10, her mother volunteered at a crisis centre where women who had been "beaten and abused, probably for years, showed up with children and the T-shirt on their back". She would help her mother at Housing Works, an organisation providing housing for families and homeless people living with HIV/AIDS. "One person had been living like a hermit and didn't have any family, any friends, and died. So here we were cleaning it out, and trying to make it nice and new again, so we could bring in someone else. It was heavy work."

Her parents and other squat residents built a stoop to keep away drug dealers, and it was hanging out there one day that she was discovered by Clark and Korine. "This guy was like, 'You just look so perfect.' And I thought, 'what are you talking about?' Harmony was hopping up and down, he was 19, and Larry can come off a little lasciviously, so I was like, 'Um, Daddy, there are random people here who have asked me to be in a movie.'"

Her father rode her to the audition on the crossbar of his bicycle. "I remember thinking, 'Oh, this looks legitimate.' It was a big office. I had to read, and Larry said, 'Is that your boyfriend outside?' And I was like, 'Ew, that's my Dad! What is wrong with you?'"

The family went on holiday to Texas with the money she made, and ended up living there. Dawson wasn't completely sold on acting then. She'd always loved maths, and started to love biology. But she ended up moving back to Manhattan, and attending the Lee Strasberg Theatre and Film Institute.

Dawson has a mixed heritage – Puerto Rican, Afro-Cuban, Irish and Native American – and says this has been an unexpected asset. "I remember having a conversation with an actress who was blonde and blue-eyed, and she was like, 'You're going to do really well here [Hollywood].' And I was still really struggling, and said, 'O-K.' And she said, 'no, Rosario, there are a million girls who show up in Hollywood every day who look like me. There's not a lot of people who look like you."

Spending time with Dawson is uplifting. Her political discussion flows from Voto Latino, the organisation she co-founded in 2004 to encourage Latino people to vote; her passionate support for One Billion Rising, Ensler's upcoming march to end violence against women; ecological campaigns; a call for an end to lobbying in Washington. She has been shooting Trance, an art-heist film directed by Danny Boyle, and is rapturously excited to be playing US labour rights activist, Dolores Huerta, in a film directed by Diego Luna.

And she talks with just as much effusive energy about the women's benefit. "I love this piece," she says, "because it's really in your face, and sometimes you've got to make people a little uncomfortable.

"There are horror movies that are made, but those are fake horrors – there are plenty of real things to be scared about, and to want to do something about. I'm just grateful," she says, like a true comic-book enthusiast, "to be able to use my powers for good, not evil".

Rosario Dawson performs A Memory, A Monologue, A Rant and A Prayer at the Lyric Theatre, London, on 26 March 2012. All proceeds go to V-Day and Women for Women International. Details: nimaxtheatres.com or 0844 482 9674.